OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
considered a stock-dependent demand model with variable holding costs, and assumed that the unit holding cost
is a nonlinear continuous function of the time the item is in stock or a nonlinear continuous function of the
inventory level. Giri and Chaudhuri [22] extended this model to account for perishable products. Roy [23]
developed an inventory model for deteriorating items with time varying holding cost and demand is price
dependent. Mishra and Singh [24] developed the inventory model for deteriorating items with time dependent
linear demand and holding cost.
In these inventory models for deteriorating item with variable holding cost assumption, demonstration of holding
cost per unit time is not closely satisfied by real aspects. So, to extend and to introduce more general holding
cost function, we have presented an Economic Quantity Model (EOQ) for decaying items. To give better fitness
to this model, we introduce deterioration governed variable holding cost. The rate of deterioration is considered
as Weibull distributed. The demand of item is assumed in power pattern. Shortages are allowed and partially
backlogged at next replenishment length dependent rate. Finally, numerical example is presented to demonstrate
the developed model and sensitivity analysis is also provided.
2 Notations
The following notation is used throughout the paper:
I()
t The inventory level at any time t , t ≥ 0 ;
T Constant prescribed scheduling period or cycle length (time units);
I Maximum inventory level at the start of a cycle (units);
max
S Maximum amount of demand backlogged per cycle (units);
t Duration of inventory cycle when there is positive inventory;
1
Q Order quantity (units/cycle);
c Cost of the inventory items ($);
1
c Fixed cost per order ($/order);
2
c Shortage cost per unit back-ordered per unit time ($/unit/unit time);
3
c Opportunity cost due to lost sales ($/unit);
4
*
ATC ( t ) Average total cost per cycle.
1
3 Assumptions
In developing the mathematical model of the inventory system, the following assumptions are made:
1. The inventory system involves only one item and the cycle length is given and finite.
2. The replenishment occurs instantaneously at an infinite rate.
3. Lead time is negligible.
4. The distribution of time until deterioration of the item follows a two-parameter Weibull distribution.
5. Deterioration occurs as soon as items are received in to inventory.
6. There is no replacement or repair of deteriorating items during the period under consideration;A brief
introduction to the rate of deterioration is given as follows: t product life (time to deterioration), t > 0; f () t
probability density function of product life (p.d.f.); F()
t cumulative distribution function of product life
(c.d.f); R()
t reliability (probability of survivorship by time t ); Z()
t instantaneous rate of deterioration. From
reliability theory, one has
R() t = 1 − F()
t , (1)
f () t
Z()
t = , (2)
R()
t
and R (0) = 1.
If the product life t is assumed to follow a two-parameter Weibul distribution, its p.d.f. f () t is
β 1 αt
β
f () t αβt − e
−
= , (3)
where α is the scale parameter, α > 0 , and β the shape parameter, β > 0 , using the former definition, one has
β
− αt
R() t = 1 − F()
t = e . (4)
Substituting Eqs. (3) and (4) into Eq. (2), one has
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